FIRST PASSAGE PROCESSES IN QUEUING SYSTEM MX / Gr / 1 WITH SERVICE

This article deals with a general single-server bulk queueing system with a server waiting until the queue will reach level before it starts processing customers. If at least r customers are available the server takes a batch of the fixed size of units for service. The input stream is assumed to be a compound Poisson process modulated by a semi-Markov process and with a multilevel control of service time. The authors evaluate the steady state probabilities of the queueing processes with discrete and continuous time parameter preliminarily establishing necessary and sufficient conditions for the ergodicity of the processes. The authors use the recent results on the first excess level processes to explicitly find all characteristics of the named processes. Some characteristics of the input process, service cycle, intensity of the system, and both idle and busy periods are also found. The results obtained in the article are illustrated by numerous examples.


systelns.
As it appeared from different sources, the evaluation of the steady state probability distribution for an emb'dded queueing process has a problem of dealing with roots of certain a analytic function which were difficult to find and whose existence were firmly related to a finite set of unknown probabilities.Although some recipes existed in the literature prior to this vork but there arC' only those al)ln'opriate which dealt with very special systems.A recent work by Abolnikov and Dukhovny [2] enabled not only to supplenent and refine the existing results but also to enlarge the class of systems to which this analysis can successfully be applied.
Dshalalow and Russell [7] were probably the first to employ the new techniques to a class of systems of type M/G/1 generalizing usual assumptions about this system by allowing the input stream to be modulated by a semi-Markov process and by' implementing a service control.This work generalized models considered by various authors (see for instance, Chaudhry and Templeton [3,4]).By assuming that, in addition, the input stream is general bulk, we obviously have another problem of the cmtical behavior of the queueing process about level r.Indeed, in this case, when the server will start his service for the first time after being idle, the queue length is more likely to exceed than to exactly reach level r.The authors apply the recent results of the first passage problem obtained in Abolnikov and Dshalalow [1] to study the behavior of the queueing process at the instant of the first passage time of level r by the process.Thus, one of the central problems in the analysis of such queueing systems is another (auxiliary) process embedded in the queueing process over the successive instants off first passage times.
The authors establish necessary and sufficient conditions for the ergodicity of the queueing process with discrete and continuous time parameters and study its steady state behavior in both cases.Due to the queue length dependent service delay discipline assumed in this article, an auxiliary random process describing the value of the first excess of the queue length above level r-appears to be one of the kernel components in the analysis of the queueing process.The authors study this process independent of its relation with the queueing system and obtain formulas for its distribution.Using these formulas the invariant probability measure of the embedded process is found in terms of generating functions and roots of a certain associated function in the unit disc of the complex plane.
The stationary distribution of the queueing process with continuous time parameter is derived by using semi-regenerative techniques.The authors also obtain the intensity of the input process, service cycle, a formula for the capacity of the system, the distribution of an idle period and the mean busy period.A number of various examples illustrate the results obtained in the article.

AN INFORMAL DESCRIPTION OF THE SYSTEM
Let Q(t) give the total number of customers in the system at time t, and let the stochastic sequence {T,,; n 0,1,...; T O 0} gives the successive instants of time when the server completes his service.Consider the embedded sequence {Q(T, + 0) n 0,1,...} which gives the total number of the customers in the system immediately after a batch of processed units departs from the s)t'm.If at tme T, + 0 the queue length Q,, is greater than or equal to r, the server takes any group from the' queue of a fixed size r and begin processing this group in accordance with an arbitrary distribution function, B,,, generally dependent on Q,.If Q,, < r the server rests as long as the queue needs to accumulate its level to at least r.The server activity is fully restored at the instant of time, say if,,, the queue length reaches or exceed level r. [As mentioned in the introduction, because customers arrive in groups the queue level at time ft, is more likely to exceed r than to reach it.] In addition we assume that in the interval (T,, T,,+ ] the input flow is a compound Poisson process with parameter t(Q,,).This assumption allows a greater flexibility of the system incorporating a natural reaction of the input flow on the state of the system.
In the next section we will study the behavior of another embedded process Q(,).We will be using some basic results on the first passage problem stated and developed in Abolnikov and Dshalalow [1].

PRELIMINARIES ON THE FIRST PASSAGE TIME PROBLEM PROCESSES
First we treat the process {Q(ff,)} without any connection to the queueing system.Because of its "conditionally-regenerative" properties we will study the point process ft, which can obviously be described by a certain auxiliary integer-valued renewal process S {S X 0 + X + + X, n 0,1,...} whose successive increments X,X,..., give sizes of groups of the input process arriving at the system.Unlike the (usually regular) renewal process S itself, the concrete process Q(ff,,) which S describes, is terminated for some n.
In this section we will discuss the "critical behavior" of a compound Poisson process Z determined by a Poisson process r {r,, o + t + + t, ;n > O, o O} on lit + marked by a discrete-valued delayed renewal process S {S,, X o + X1 + + X,, ;n > O} on {0,1,...}.
We assume that the processes r and S are independent.We also assume that inter-renewal times t,, %-r,,_, are described in terms of its common Laplace-Stieltjes transform e(O) E[e-'"] A--' n 1,2,....For convenience we agree to set ft, T,, as long as X0 Q, _> r.
For a fixed integer r _> we will be interested in the behavior o the process S and some related processes about level r.
The following terminology is introduced and will be used throughout the paper.
(i) For each n the random variable u,, inf{k >_ O" S >_ r} is called the indez of the the first ezcess (above level r-1).
Ve formulate the main theorems from Abolnikov and Dshalalow [1] and give formulas She joint distributions of the first passage time and the random variables listed in 3.1 3.2 THEOREI.The f7c*wnal 7'(0.z) (o He fra* pa..saW *me and of Oze mde o *he ('(e,z) Specifically, the Laplace-Stieltjes transform of the first passage time, 0)(0,1), is follows: ikr.
From formula (3.2a) we immediately obtain that the mean value o[ the index o[ the first excess equs 0, ir.
Prom (a.2a) we also obtain the mean value of the first psage time: a.5 THEOREM.The eerator G,(O,) of the first ecess level can e deteined om the following formula: The rationale behind the use of the term "generator of the first excess level" comes from the following main result.
3.6 THEOREM.The functional (](')(0,z) (of the first passage time and of the first ezcess level) can be determined from the formula 3.7 REMARK.To obtain the functionals of the marginal processes defined in (3.1b-3.1(1)we set e(O)= in formulas (3.2a), (3.5a) and (3.6a).
3.10 REMARK.Now we notice that the above results can be applied to our queueing system, where in formulas (3.2a)-(3.9a)we supply a(z) with subscript i.

PROJECTIVE OPERATORS TECHNIQUES
In this section a formula for the generating function of the process {S,} will be derived in another form.
Let f be an analytic function in the annulus A(0,0,1) {z e C: 0 < z < x} d conti- nuous on the boundary of the unit disc, F {ll z 1}.Then f equals its Laurent series f for all points z E A(0,0,1).Denote T+f the tame part and T-f the principal part of that Laurent series.We mention a few properties of the operators T + and T-" [Pb] Let X be a random variable valued in Il 0 such that E[zx] =a(z) ad let A {0,1,...,r-1}.Denote A= {r, r + 1,...}.Then from IF4] it follows that (P) )   where I A denotes the indicator function of set A and the operator (f)+ denotes {y,0}.
Hence we obtain fi'om [P1] that Now the statement of the corollary follows fi'om (4.2c) and properties [P1-P4].

FORMAL DESCRIPTION OF THE SYSTEM
We begin this section with definition a modulated process introduced and studied in Dshalalow [6].All stochastic processes below will be considered on a probability space {f,q,(P*),, }, with ty {0,1,...}.
Let {T,,; n 0,1,...; T o 0} be a point process and let (t) be an integer-valued jump process with successive jumps at T,, (we allow (T,,)= (T,, + ) with positive probability).Let {r,; k E N} be a non-stationary orderly Poisson point process with intensity function A(t).
INPUT.Let C(-) be the counting measure associated with the point process {T,,}.
Assume that the input is a compound Poisson process nodulated by {((t)}, where is ith batch size of the input flow which depends on ((t).Thu, in our case {X} is an integer- valued doubl.stochasticprocess describing the sizes of groups of entering units.We denote ae{,)(z) Eli ,e,}], i= 1,2 the generating function of ith component of the process {X}.SERVICE TIME AND SERVICE DISCIPLINE.At time T, +0 the server takes a batch of units of size r from the queue and serves it during a random length of time a,+ if the queue length Q,, is at least r.Otherwise, the server idles until the queue length for the first time reaches or exceeds the level v. Let ,, mf{k 6 : r T,,}, 6 0. Then the size of the first group Mter T, (which arrives at the instant of time r e.) is X,,,.For a convenience in notation, we reset the first index-counter of the process {X} on after time hits T,.
Therefore, in the light of the new notation, X,,X,,... will denote the sizes of successive groups of units arriving at the system pt time T,.Let S, XoQ, + X, + + X,, where Xo,=Q,.Then, given Q., {S.; k6} is an integer-valued delayed renewM process.Denote u, inf{k O" S, r} the rdom index when the process {S,} first cs o exceeds level r given that the queue length is Q,.If Q, r, T, + T, coincides with length of service time a, + of the n + 1st batch.If Q, < r the interval (T,, T, + ] contains the waiting time for XQ, + + Xu,, units to arrive and the actual service time a, + .In both ces we sume that a, + h a probability distribution function B, fi {B0, B ,...}, where the latter is a given sequence of arbitrary distribution functions with finite means {b0, b ,...}. Finally, denoting V, Z(a,) we obtMn the following relation for process s.-+ v.+,, .<r 6. EMBEDDED PROCESS From relation (6.1) and the nature of the input process it follows that the process {,q,(PX)x, Q(t);t >_ 0} @ has at T,, n _> 1, the locally strong Markov property (see defini- tion A.3 in Appendix) and that {,5,(P')x, Q,;n.lo} is a homogeneous Markov chain with transition probability matrix A (a,1).Let A(z) denotes the generating function of ith row of matrix A. Since A(z) E[z] we obtain from (6.1) that (6.1) A,(z) g,(z)z-')(z), c:., where (6.1a) 9,(z) 3,(,-A.a,(z)),3,(0), Re(O) >_ O, is the Laplace-Stieltjes transform of the probability distribution function B,.
Therefore, given that p< r, the Markov chain {Q,} is ergodic.Let P=(p;x ) be the invariant probability measure of operator A and let P(z) be the generating function of the components of vector P. Denote ' + {z C: z _< }.Now we formulate the main result of this section.
6.4 LEMMA.The ezpected number of units -'}= I. '1-that arrive dumn9 an idle pemod (of the server) started with the queue length equals (defined in formula (6.3e)) can be represented in the form (6.4a) ')= a,! PROOF.Lemma 6.4 is just another interpretation of formula (3.9b).
f'! 6.5 REMARK.Consider the following auxiliary process.Let {12, Y,(P*),,,, Y,, n I} 0 (abbreviated {Y}) be a homogeneous irreducible and aperiodic Markov chain with transition probability matrix A {a,; i,j } and the generating function A,(z) of ith row of A. Assume that A,(z)= zl'-l+9(z), 1, where 9 is analytic in r + and continuous on F.
,mO iz ,_+ R(z) |'-0 z () According to theorem A.2 (in Appendix), the function z-, z"-g(z) has exactly r roots (counted with their multiplicities) in '+.Since the function in the left-hand side of (6.5c) is analytic in F + and continuous on F the polynomial R(z) should have the same r roots (given that gt(1) < r).Taking into account that the left-hand side of (6.5c) equals for z 1, we get The polynomial R, which determines the generating function 5(z), can uniquely be restored after we find all roots of z-9(z) in + and satisfy (6.5d).In general, the roots can be evaluated numerically (in some cases also analytically).Alternatively, the roots, and therefore the polynomial R, can be obtained from computer simulation of the Markov chain {Y}.

APPLICATIONS AND EXAMPLES
We now will consider several special cases of our system.All of them can be analyzed with the help of the general theorem 6.3 and by using appropriate numerical methods.However, in the particular cases considered below, it turns out to be possible to develop a more direct approach and thereby obtain more convenient formulas for generating functions of the corresponding queueing processes.In the first case we drop the modulation of the input process and service control, thereby assuming that A,= A, a,(z)=a(z), and B,= B,i'.The folloving proposition states that in this case the generating function P(z) can be expressed in terms of the stationary prol)abilities for the auxiliary Markov chain {Y} introduced in 6.5.
7.1 PROPOSITION.Gwen that p < r, tte generating function P(z) of the invariant probab.ditymeaz'ure P of operator A for the embedded process Q,,} in a bulk queueing system wdh queue length, dependent sewzce delay and w,thout modulation and e'wce control, can be determined from the followzng formula: where ((z) is found by formula (6.1) and R(z) is defined in (6.2).
PROOF.In the assumptions of the proposition, formula (6.3a) for P(z) reduces to 1.. (7.15) The first factor z g(z)[1-a(z)] a(1z) in (6.6c) has the following obvious properties: It is analy-tic in F + and continuous on F, it is valued at z and it does not vanish at all roots of zg(z) inside +.The latter property and analyticity of P(z) yield that the function r-, p,a,( (z-1)E, 0 is analytic in F + and continuous on F and it takes on value at z 1.Thus, the numerator of (7.1d) must have the same roots with their multiplicities as the denominator in F + and it assumes value r-g(1) at z 1.Since G,,(z) is a (r-1)th degree polynomial the numerator of (7.1d) is a rth degree polynomial.Therefore, from the considerations in 6.5 it follows that the numerator of (7.1d) must be equal to R(z) defined in (6.55).

Fi
In the second example we assume that B,(x) f,B(x), < r, S,(x) B(x), >_ r, and A, .k,a,(z)= a(z), i .L et o() denote the Laplace-Stieltjes transform of the (arbitrary) pro- bability distribution function F and let H(z) V(A-Aa(z)).
7.2 PROPOSITION.Under the above assumptions, the generating function P(z) of the invariant probability measure P of operator A is determined by the expression a(z) L(z)R(z) (7.2a) P(z) g(z) a(l z) z 9(z)   where the function L is an analytic in r + and continuous on F, and L(1) 1.
8. ANALYSIS OF THE QUEUEING PROCESS WITH CONTINUOUS TIME PARAMETER In this section our main objective is the stationary distribution of the queueing process with continuous time parameter.
(i) Let fl= E[T]] (the mean sojourn time of the process {(t)} in state {j}) and fl (; j )T.Then we will call the value Pf/the mean service cycle of the system, where P denotes the stationary probability distribution vector of the embedded queueing process {Q.}.
(ii) Let a= (tr;z q @)T, A= (A;x q I') T and let p= a, fl,, be the Hadamard (entry- wise) product of vectors a, fl and ,.We call the scalar product Pp the intensity of the system.
Observe that the notion of the "intensity of the system" (frequently called the offered load in queueing theory) goes back to the classical M/G/1 system, when Pp reduces to p Ab.
It is noteworthy that in many systems the intensity of the system and the server capacity coin- cide.
PROOF.Recall that the Markov renewal process {Q,T} is ergodic if p<r.By corollary A.8 the semi-regenerative process {()} has a unique stationary distribution provided that p < r.From (8.5a) we can see that the semi-regenerative kernel is Riemann integrable over N+.Thus, following corollary A.8 we need to find the integrated send- regenerative kernel H (which is done with routine calculus) and then generating functions h(z) of all rows of H. Then it follows that where G'I(z) denotes the tail of the generating function G,(:) summing its terms from r to .However, it is easy to show that G;I(z) and G,(z) coincide.Then it appears that where the index j can be dropped for all j exceeding N, in accordance with sumption (AS2) made in section 4. ormula (8.6a) now follows from corollary A.8, equations (8.6c-8.6e),(a.6a), remk a.10, and some algebraic transformations.
The following corollary (which follows from (8.aa), (8.4a), (8.2a) and (8.5a) by mes routine calculus) gives an elegant formula for the service cycle P and the generating function (z) if we just drop the modulation of the input but retain bulks of the input, service control d state dependent service delay. 8.7 COROLLARY.The seice ccle P nd the 9enertm9 function () of in the qeein9 sstem th no modulation of the input cn be detemed from the followin9 (s.7) =.
[-(1 8.8 EXAMPLES. (i) Observe that the same result as (8.7a) holds true by retaining a "weak modulation", i.e. assuming that A, A and a, a but having no further restriction to the generating functions a,(z).
(ii) Assuming that the input is an orderly modulated Poisson process, in other words if aj(z) z, but retaining all other assumptions we arrive at the result by Dshalalow and Russell  [7].Indeed, hj is reduced to hs(z)-(l z)z- (A.8a) r(z) E ,, P h'j( z) PH Finally, PROOF.From (A.7a) we get an equivalent formula in matrix form, r formula (A.8a) is the result of clcnmntary algebraic transformations.13 A.9 THEOREM (Dshalalow [6]).Let Z be a compound Potsson process modulated by a semz-Markov proccs, tn accordance with the above notatto'o ad deftntton tn section 5. Let p a.fl.denote the Hadamard product of vectors a, fl and A. If {Q,,, T,,} is ergod'tc t/ten the intensity of the process Z is given by the formula -) E'[e %:%], (]1')(8.-)E'[c %= %].